We report a direct measurement of the low-frequency noise spectrum in a superconducting flux qubit. Our method uses the noise sensitivity of a free-induction Ramsey interference experiment, comprising free evolution in the presence of noise for a fixed period of time followed by single-shot qubit-state measurement. Repeating this procedure enables Fourier-transform noise spectroscopy with access to frequencies up to the achievable repetition rate, a regime relevant to dephasing in ensemble-averaged time-domain measurements such as Ramsey interferometry. Rotating the qubit's quantization axis allows us to measure two types of noise: effective flux noise and effective critical-current or charge noise. For both noise sources, we observe that the very same 1/f-type power laws measured at considerably higher frequencies (0.2-20 MHz) are consistent with the noise in the 0.01-100-Hz range measured here. We find no evidence of temperature dependence of the noises over 65-200 mK, and also no evidence of time-domain correlations between the two noises. These methods and results are pertinent to the dephasing of all superconducting qubits. © 2012 American Physical Society.